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Sharp defective log-Sobolev inequalities on H-type groups

Gioacchino Antonelli, Mattia Calzi, Maria Gordina

Abstract

In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.

Sharp defective log-Sobolev inequalities on H-type groups

Abstract

In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.

Paper Structure

This paper contains 13 sections, 13 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Statements of the results
  3. Preliminaries
  4. H-type groups and hypoelliptic heat kernels
  5. Sobolev inequalities on $H$-type groups
  6. BE inequality and log-Sobolev inequality in CD spaces
  7. Proof of \ref{['thm:NonAbelianHtype']}
  8. Estimates on the heat kernel
  9. Exponential integrability of Lipschitz functions
  10. A defective log-Sobolev inequality for the Gaussian-like measure with respect to the sub-Riemannian distance
  11. Proof of \ref{['thm:Main1']}
  12. A technical Lemma
  13. More general $2$-step groups

Key Result

Theorem 1.1

Let $G$ be an $H$-type group. Define Hence and

Theorems & Definitions (32)

  • Definition 1
  • Remark 1
  • Theorem 1.1
  • Remark 2
  • Corollary 1: Fernique's Theorem
  • Theorem 2.1: JerisonLee1988YangQ2024
  • Remark 3
  • Corollary 2
  • proof
  • Remark 4
  • ...and 22 more